Assessing the (a)symmetry of concentration-effect curves

Pharmacology & Therapeutics 95 (2002) 21 – 45

Assessing the (a)symmetry of concentration-effect curves: empirical versus mechanistic models Jesu´s Giraldoa,*, Nuria M. Vivasb, Elisabet Vilab, Albert Badiab a

Laboratori de Medicina Computacional, Unitat de Bioestadı´stica, Facultat de Medicina, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Spain Departament de Farmacologia, de Terape`utica i de Toxicologia, Facultat de Medicina, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Spain

b

Abstract Modeling the shape of concentration-effect curves is of prime importance in pharmacology. Geometric descriptors characterizing these curves (the upper and lower asymptotes, the mid-point, the mid-point slope, and the point of inflection) are used for drug comparison or for assessing the change in agonist function after a system modification. The symmetry or asymmetry around the mid-point of a concentrationeffect curve is a fundamental property that, regretfully, is often overlooked because, generally, models yielding exclusively symmetric curves are used. In the present review, empirical and mechanistic models are examined in their ability to fit experimental data. The geometric parameters of a survey of empirical models, the Hill equation, a logistic variant that we call the modified Hill equation, the Richards function, and the Gompertz model are determined. To analyze the relationship between asymmetry and mechanism, some examples from the ionic channel field, in an increasing degree of complexity, are used. It is shown that asymmetry arises from ionic channels with multiple binding sites that are partly occupied. The operational model of agonism is discussed both in its empirical general formulation and including the signal transduction mechanisms through G-protein-coupled receptors. It is shown that asymmetry results from systems where receptor distribution is allowed. Developed mathematical models are compared for describing experimental data on a-adrenoceptors. The existence or not of a relationship between the shape of the curves and receptor reserve is discussed. D 2002 Elsevier Science Inc. All rights reserved. Keywords: Ionic channels; G-protein-coupled receptors; Receptor reserve; Curve fitting; Slope parameter; Asymmetry Abbreviations: [A], agonist concentration; CI, confidence interval; E, pharmacological effect; GPCR, G-protein-coupled receptor; log, logarithm to base 10; nH50, the Hill coefficient at the mid-point; SHR, spontaneously hypertensive rats; x, log[A]; xI, point of inflection; x50, log[A] giving 50% of the maximum effect; xb, parameter present in some empirical models (Hill, Richards, Modified Hill) that contributes most to the location of the E/x curve along the abscissa axis; if the curve is symmetric, it coincides with both xI and x50; WKY, Wystar Kyoto.

Contents 1. 2.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The Hill equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The Richards function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The Gompertz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The modified Hill equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Comparison of the models. . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Ionic channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. A receptor with one binding site . . . . . . . . . . . . . . . . . . . . 3.1.2. A receptor with n bindings sites either vacant or fully occupied . . . . 3.1.3. A receptor with two binding sites where partial occupation is allowed

* Corresponding author. Tel.: +34-93-5812348; fax: +34-93-5812344. E-mail address: [email protected] (J. Giraldo). 0163-7258/02/$ – see front matter D 2002 Elsevier Science Inc. All rights reserved. PII: S 0 1 6 3 - 7 2 5 8 ( 0 2 ) 0 0 2 2 3 - 1

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A hybrid model: the operational model of agonism. . . . . . . . . . . . . . . . . . . . . 4.1. The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Signal transduction by G-protein-coupled receptors: the stoichiometric relationship between the receptor and the G-protein . . . . . . . . . . . . . . . . . . . . . . . 5. Asymmetry and receptor reserve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The pharmacological systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. System 1. Contractile response to norepinephrine in a1-adrenoceptor vas deferens from spontaneously hypertensive and Wystar-Kyoto rats . . . . . 5.1.2. System 2. Contractile response to phenylephrine in endothelium-denuded aorta rings from spontaneously hypertensive and Wystar-Kyoto rats . . . . 5.2. Pharmacological analysis: the operational model of agonism . . . . . . . . . . . . 5.2.1. Application to System 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Application to System 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Pharmacological analysis: the logistic fitting procedures . . . . . . . . . . . . . . 6. Are the Hill coefficient and receptor reserve two related properties? . . . . . . . . . . . . 7. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Computational details, data analysis, and software . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction The interaction between drugs and receptors is usually studied in isolated tissues, where the applied agonist concentration [A] links to the observed physiological effect, allowing the construction of [A]-effect (E/[A]) curves. Design and analysis of E/[A] curves is a central issue in pharmacology. Mathematical equations developed for modeling E/[A] data can be useful tools for the understanding of physiological function, for classifying receptors and ligands, and for designing more selective and potent therapeutic agents. Because all measures of drug activity or system sensitivity come from E/[A] curves, it is of fundamental importance that the mathematical models fit the experimental data correctly. In this review we present a survey of empirical and mechanistic approaches to quantitatively analyze E/[A] data. E/[A] curves are commonly depicted in a semi-logarithmic scale, E/x, where x = log[A]. Typically, E/x data follow sigmoid curves that are bounded by two horizontal asymptotes: the lower asymptote (E = 0, in the absence of basal response, which for simplicity will be the case throughout this review) and the upper asymptote (E = a). To characterize an E/x curve, four geometric descriptors are used: the upper asymptote (maximum response), the mid-point (curve location), the mid-point slope (steepness), and the point of inflection (symmetry of the curve). Each of these properties has a pharmacological meaning and can be mathematically formulated. The upper asymptote reflects the intrinsic activity of an agonist and is defined as the value toward which the effect approaches as the [A] increases, a ¼ limx!1 E. The mid-point, x50, dictates the potency of a drug and is defined as the logarithm of the [A] for half the value of a. The mid-point slope is the value of the slope of the E/x curve at the mid-point, ((dE)/(dx))x = x50, and displays the

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sensitivity of the system to small changes in [A]. Rectangular hyperbolic curves give a typical mid-point slope of 0.576 when they are normalized (the derivative is divided by a), while non-hyperbolic curves can be steep (dE/dx)x = x50 > 0.576) or flat (dE/dx)x = x50 < 0.576). The point of inflection, xI, is a point on a curve at which the curvature changes from convex to concave or vice versa. For an E/x curve, this is a point at which the first derivative of the function is a maximum, whereas the second derivative is equal to 0. Besides, the location of the point of inflection serves for the assessment of the symmetry of the curve. An E/x curve is symmetric if the point of inflection matches the mid-point, xI = x50, and asymmetric if it does not, xI 6¼ x50 (Fig. 1). Whereas analysis of the asymptote, the mid-point, and the mid-point slope are commonly met in pharmacological literature, the point of inflection and so, the presence or absence of asymmetry is frequently missed. In our opinion, this problem is of enough importance to merit special consideration. First, the shape of theoretical E/x curves and their geometric descriptors can be very dependent on including or not the possibility of asymmetry. It has been shown (Van Der Graaf & Schoemaker, 1999) that analysis of E/x curves with models that do not account for asymmetry could lead to wrong estimates of the asymptote, location, and steepness parameters. Thus, it seems reasonable that if experimental data points appear to follow an asymmetric curve, the use of a theoretical model accounting for asymmetry would be a pertinent decision. Second, it would be expected that quantification of asymmetry could be worthwhile for a better characterization of the pharmacological pathways and the mechanisms of signal transduction, although a word of caution is needed because too little is known about stimulus-response coupling mechanisms. In this regard, it recently was postulated that receptor reserve

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Second, some examples taken from the ionic channel field, the paradigm for mechanistic models, will be discussed. Third, the operational model of agonism will be reviewed. The operational model of agonism is an empirical model, although it can be reformulated to include in its equations particular mechanisms. It will be proved that the operational model of agonism can account for asymmetric curves. Moreover, extensions of the operational model to signal transduction by G-protein-coupled receptors (GPCRs) will be developed and handled in a similar fashion. The proposed relationship between the shape of the curves and the receptor reserve will be discussed. Also, we will further compare the ability of some of these models to accurately describe experimental data produced in our laboratory. The discussion of pharmacological examples here may facilitate the understanding of the proposed mathematical models, and may show that these models are truly relevant. Finally, an appendix is included to facilitate for the readers the derivation of functions and parameters.

2. Empirical models Fig. 1. Qualitative comparison of symmetric versus asymmetric concentration-effect (E/x) curves. The asymptotic maximum effect has been set at E = 1. The mid-point, x50, is characterized by E = 1/2. The point of inflection, xI, is characterized by a maximum of the first derivative of the effect and a value of 0 for the second derivative. a: Symmetric concentration-effect curve: xI = x50. b: Asymmetric concentration-effect curve: xI 6¼ x50.

introduces asymmetry into E/x curves (Agneter et al., 1997). These authors proposed that a touchstone for the existence of spare receptors could be a slope parameter > 1 (steep nonhyperbolic curves) of a logistically fitted curve. Furthermore, the operational model (Black & Leff, 1983), a classical methodology to assess pharmacological agonism, was considered inappropriate to quantify receptor reserve because, supposedly, this model is inherently symmetric (Agneter et al., 1997). This issue was also the subject of a debate (Agneter et al., 1998; Giraldo, 1998). More recently, a fitting procedure to describe the degree of asymmetry of E/ x curves was presented (Van Der Graaf & Schoemaker, 1999), allowing the authors to infer conclusions about drugreceptor interactions that cannot be obtained by models that require E/x curves to be symmetrical. We see from the foregoing discussion the necessity of revisiting the problem of asymmetry. To this goal, the present review aims to provide the essential mathematics necessary for the quantitative analysis of concentrationeffect curves. Our purpose is to offer investigators a comprehensive methodology to assess the asymmetry of E/[A] curves in different pharmacological scenarios. First, a number of empirical models will be revisited. Starting from the symmetric Hill equation, asymmetric extensions, such as the Richards function, the Gompertz model, and a logistic variant of the Hill equation, will be examined in detail.

Empirical models are used to find the mathematical equation that best fits experimental data. They can be applied to properly summarize numerically and graphically the response of a pharmacological system to [A]. Although the equation parameters of empirical models lack physical meaning, they can be very useful to quantitatively assess physiological function. The geometric parameters characterizing the E/x curves (location, asymptote, mid-point slope, and point of inflection) can be used to compare the potency and efficacy of different agonists and to investigate the effects of experimental interventions at either the receptor or the transducer level. 2.1. The Hill equation The Hill equation is, despite its limitations, the most frequently used model in pharmacology to analyze E/[A] curves. This model fits E/x data by using the equation E¼

a 1 þ 10mðxb xÞ

ð1Þ

where x = log[A] and m > 0, m being the Hill coefficient (Section A.1). The upper asymptote is a, the mid-point is x50 = xb, the mid-point slope is (dE/dx) x = x 50 = (amln 10)/4 = 0.576am, and the point of inflection is xI = xb. The point of inflection matches the mid-point, irrespective of the values of the model parameters. Thus, the Hill function gives rise to symmetric curves in all cases and, therefore, is not a suitable method to model experimental asymmetric E/x curves. Fig. 2 shows a simulation of E/x curves complying with Eq. (1) (Fig. 2a), together with their second derivatives (Fig. 2b) for 6 different values of the

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the value of the asymptote a, which probably is the most difficult quantity to measure accurately. If the estimate is wrong, all of the values of the dependent variable (i.e., E/ (a  E)) will be biased accordingly. Moreover, the transformation log(E/(a  E)) can produce meaningless results for E values close to 0 or the asymptote a because of the small value of the numerator or denominator and the amplification of the error by the logarithmic relationship (Taylor & Insel, 1990). For these reasons, the fit of the hyperbolic form obtained by iterating about the values of xb, m, and a is likely to differ from that of the linear fit obtained by iterating about the slope and y-intercept. 2.2. The Richards function A way to account for asymmetry while keeping the general structure of the previous model is to add a new parameter. This function (Richards, 1959) is a generalization of the Hill equation, and may be expressed in the following form: E¼

Fig. 2. Concentration-effect (E/x) curves (a) and their second derivatives (b) produced by the Hill function (Eq. (1)). Fixed parameter values in the simulation are a = 1 and xb = 0. All the E/x curves are symmetrical: the mid-point and the point of inflection take the same value (0).

slope parameter (m = 0.6 and 0.8, flattened hyperbolae; m = 1, rectangular hyperbola, and m = 1.2, 2, and 4, steep hyperbolae) and common asymptotic (a = 1) and location (xb = 0) parameters. The slopes of the curves at the midpoint (x50 = xb = 0) are 0.345, 0.461, 0.576, 0.691, 1.151, and 2.303 for m = 0.6, 0.8, 1, 1.2, 2, and 4, respectively. Fig. 2a clearly depicts the increase in the steepness of the curves when increasing the m value. However, in all cases, the E/x curves are symmetric because their second derivatives at the mid-point are equal to 0 (Fig. 2b). The Hill plot, which is constructed as log (E/(a  E)) = mx  mxb, where a is the maximum response, is a transformation of Eq. (1) that can be useful for the identification of asymmetric E/x curves. The theoretical Hill plot yields straight lines of slope m. Because the Hill plot and the Hill equation are essentially equivalent models, how well the Hill plot fits (straight lines) would be an indicator of how well the Hill equation fits (symmetric sigmoid curves) and vice versa. Thus, the data points depicted as log (E/ (a  E)) against x = log[A] would lead us to the conclusion that the former E/x is a symmetric or asymmetric curve if data are well or badly aligned along a straight line, respectively. In the latter case, alternative models to the Hill equation would be needed. Some caution is indicated about the use of the Hill plot. First, to construct the plot, the investigator needs to know

a ð1 þ 10mðxb xÞ Þs

ð2Þ

where s > 0 (Section A.2). The upper asymptote is a, the mid-point is x50 = xb  (1/m) log(21/s  1), the mid-point slope is (dE/dx)x = x50 = 1 ðða  m  ln 10  s  ð1  21=s ÞÞ=2Þ, and the point of inflection is xI = xb + (1/m)log s. The new parameter, s, allows for asymmetry. If s = 1, Eq. (2) is equivalent to Eq. (1) and the theoretical curve is symmetric. Consistently with this feature, we see that if s = 1, then xI = x50 = xb. However, if s 6¼ 1, then xI 6¼ x50 and the theoretical curve is asymmetric. Interestingly, for s 6¼ 1, the degree of asymmetry of the curve, measured as the difference between xI and x50, relies on both s and m parameters, and xI  x50 = (1/m)log (s(21/s  1)). If s > 1, then xI < x50 and the point of inflection goes before the mid-point, whereas if s < 1, then xI > x50 and the point of inflection goes after the mid-point. Fig. 3 displays the variation of xI  x50 in relation to s within the range [0.2, 2] for 3 m values (0.5, 1, and 2). The degree of asymmetry is higher in the region where s < 1 than in the region where s > 1. This is because the term (s(21/s  1)) tends to a constant value (ln 2) when s ! 1, whereas the same term tends to + 1 when s ! 0. Furthermore, for a given s, xI approaches x50, and asymmetry vanishes as m increases. The E at the point of inflection is E(x = xI) = a/(1 + 1/s)s, which depends on s, but not on m, parameter. For s = 1, E(x = xI) = a/2, as expected for a symmetric curve, but for s > 1, E(x = xI) < a/2 and for s < 1, E(x = xI) > a/2. Fig. 4 displays the E/x curves produced by the Richards function for the m and s parameters taking the values 0.5, 1, and 2. The degree of asymmetry is assessed graphically by plotting the mid-point and the point of inflection for each curve. In agreement with the above discussion, curves with s = 0.5 (s < 1) show greater asymmetry than curves

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alternative or complementary to the Hill function. Two models will be considered: the Gompertz model and the modified Hill equation. 2.3. The Gompertz model The Gompertz function (Gompertz, 1825) may be written as E¼ Fig. 3. Simulation of the degree of asymmetry, measured as the difference between the mid-point and the point of inflection of concentration-effect (E/x) curves produced by the Richards function. The asymmetry parameter s ranges between 0.2 and 2. The degree of asymmetry is 0 for s = 1, increases rapidly in the region s < 1 as s ! 0, and tends to the constant value (1/m)log (ln 2) in the region s > 1 as s ! 1.

with s = 2 (s > 1). In relation to the slopes of the curves, we see that they essentially depend on the value of the m parameter. This is due to the property that the quantities s and (1  (1/(21/s))) from the product (s(1  (1/(21/s)))) present opposite trends and in part, compensate each other. We can conclude that although both m and s parameters contribute to the slope and the asymmetry of an E/x curve, the slope is determined mainly by m, whereas the asymmetry is determined mainly by s. The Richards function (four-parameter logistic model) appears to be an improvement over the Hill equation (threeparameter logistic model) because of its greater flexibility and applicability to a larger sample space (symmetric and asymmetric curves). Regretfully, it has been found that in a number of cases, the Richards function performs deficiently in curve fitting, and individual parameter estimates are difficult to obtain (Van Der Graaf & Schoemaker, 1999). The reason for that may lay in the inclusion of the further parameter s, the one necessary for modeling the presence of asymmetry. Increasing the number of parameters of a model normally leads to an improved fit (see Section 2.5). However, addition of an extra parameter bears, as a negative counterpart, an increase of the correlation among the parameters and, accordingly, an increase of the standard errors associated with the parameter estimates, with the consequent lack of statistical significance (Freund & Litell, 1991). In our fittings, we have found a strong correlation (R > 0.9) between some of the parameters of the Richards model, mainly between xb, m, and s. This correlation can affect the reliability of the location, slope, and symmetry parameters yielding nonsensical values or very large errors. These strong correlations could also be the reason that, in many cases, the fittings with the Richards model failed to converge when applied to our data. The above discussion suggests the convenience of exploring the fitting capabilities of some other models with fewer parameters than the Richards equation as

a mðx xÞ e10 I

ð3Þ

The upper asymptote is a, the mid-point is x50 = xI  (1/m)log (ln 2), the mid-point slope is (dE/dx)x = x50 = ((amln 10ln 2)/2) = 0.798am, and the point of inflection is the parameter xI (Section A.3).

Fig. 4. Concentration-effect (E/x) curves produced by the Richards function (Eq. (2)) for (a) m = 1, (b) m = 0.5, and (c) m = 2. Fixed parameter values in the simulation are a = 1 and xb = 0. The distance between the mid-point and the point of inflection measures the asymmetry of the curve.

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Fig. 5. Concentration-effect (E/x) curves produced by the Gompertz model (Eq. (3)). Fixed parameter values in the simulation are a = 1 and xI = 0.

The Gompertz model is inherently asymmetric with xI < x50 for any value of m. The inflection point yields a fixed response value of E = a/e, lower than the response value for the mid-point, E = a/2, independently of the raw data. Fig. 5 depicts three theoretical E/x curves obeying Eq. (3). We see that all curves are asymmetric, with the point of inflection different from the mid-point. However, analogously to the Richards function, xI tends to x50 as the parameter m increases. It can be proven that the Richards function tends to the Gompertz function as the asymmetric parameter s of the Richards function increases (Section A.4). Thus, the Gompertz function could be an alternative to model experimental asymmetric curves when the Richards function fails in curve fitting.

value of the effect at the point of inflection is the same for the modified Hill equation and the Richards function because for the latter, this quantity is independent of m. We have defined the degree of asymmetry of a curve as the difference between xI and x50. It can be seen that xI  x50 = log(p(21/p  1)). The modified Hill equation is symmetric, xI = x50 = xb, only for p = 1. If p < 1, then xI > x50, whereas if p > 1, then xI < x50. Fig. 6 shows a simulation of E/x curves complying with Eq. (4) (Fig. 6a), together with their second derivatives (Fig. 6b), for 6 different values of the p parameter (p = 0.6, 0.8, 1, 1.2, 2, and 4) and common asymptotic (a = 1) and location (xb = 0) parameters. The value of p determines both the steepness and the asymmetry of the curve. The slopes of the curves at the mid-point are 0.473, 0.534, 0.576, 0.606, 0.674, and 0.737 for p = 0.6, 0.8, 1, 1.2, 2, and 4, respectively. In addition, the pair of values (x50, xI) are (  0.34,  0.22), (  0.14,  0.10), (0, 0), (0.11, 0.08), (0.38, 0.30), and (0.72, 0.60) for p = 0.6, 0.8, 1, 1.2, 2, and 4, respectively. 2.5. Comparison of the models Fitting E/[A] curves under an empirical framework bears the unavoidable question of deciding which is the best equation to use. Models initially can be compared by visual inspection of the theoretical curves that they produce superimposed on the experimental data. This subjective impression can be transformed in a quantitative measure if the models are

2.4. The modified Hill equation The Gompertz model has the drawback of its inherent asymmetry. To avoid this problem, the Hill equation can be slightly transformed while maintaining the number of parameters, 3, of the model. This alternative model, which we will call the modified Hill equation, follows the expression: E¼

a ð1 þ 10xb x Þp

ð4Þ

with p > 0 (Section A.5). The modified Hill equation is not a new model. It was first derived by Sips (1948, 1950) and the properties of the corresponding saturation function have been discussed by Boeynaems and Dumont (1980). Comparing Eq. (4) with Eq. (2), we see that the modified Hill equation is equivalent to the Richards function, with the m parameter of the Richards function fixed to 1. It can be shown that the upper asymptote is a, the midpoint is x50 = xb  log(21/p  1), the mid-point slope is (dE/dx)x = x50 = (aln10p(1  (1/21/p))/2), and the point of inflection is xI = xb + log p. These quantities are in agreement with the corresponding ones from the Richards function with m = 1. In addition, the expression for the

Fig. 6. Concentration-effect (E/x) curves (a) and their second derivatives (b) produced by the modified Hill equation (Eq. (4)). Fixed parameter values in the simulation are a = 1 and xb = 0.

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nested (one model can be formulated as a particular case of the other). In this case, the relative degree of fit can be assessed by standard test theory through an F-statistic (extra sum of squares analysis). Fig. 7 shows the hierarchical structure of the aforementioned empirical models. The Hill, the modified Hill, and the Gompertz models are nested within the Richards model, but not between them. Taking the Richards model as a reference, the Hill and the modified Hill equations are particular cases, with the parameters s or m, respectively, equal to 1; whereas the Gompertz model is the limiting case of the Richards function, as s ! 1. Increasing the number of parameters leads to a reduction of the residual sum of squares and, accordingly, to an improved fit. Thus, for the models depicted in Fig. 7, the Richards model in principle would be preferable. However, the effect on the sum of squares exerted by an additional parameter can be statistically not significant. In this case, the use of the model with an extra parameter would not be justified. The hypothesis of whether a model with more parameters significantly improves a nested model with fewer parameters is tested by the F-statistic defined as: F¼

SS1 SS2 df 1 df 2 SS2 df 2

ð5Þ

where SS is the residual sum of squares, df is the degrees of freedom, and the subscripts 1 and 2 correspond to the model with fewer and greater number of parameters, respectively. Another way to analyze the adequacy of the Richards model in comparison with aforementioned nested equations could be by assessing the values of some of the parameters of the Richards model (see Eq. (2)). If the Richards s parameter is not significantly different from 1, the Hill equation should provide an equivalent fit. An analogous

Fig. 7. Hierarchical structure of the empirical models. The Hill function, the modified Hill function, and the Gompertz model are nested within the Richards model, but not between them. Taking Eq. (2) as a reference, the Hill function matches the Richards model when the parameter s is equal to 1. The modified Hill function matches the Richards model when the parameter m is equal to 1. The intersection of the Richards, Hill, and modified Hill models corresponds to the symmetric rectangular hyperbola curve (m and s equal to 1). The Gompertz model is the limiting case of the Richards model, as the parameter s tends to infinity.

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assumption could be set for the Richards m parameter and the modified Hill equation. Also, although more subjective, a large value of the Richards s parameter would suggest the alternative use of the Gompertz model. The Hill and the modified Hill equations are not nested models, and formal comparisons between them are not easy to carry out. However, it is worthwhile to point out that although the modified Hill equation could be considered more flexible than the Hill equation because it can model asymmetric curves, this does not mean that fittings with the former will necessarily be better than with the latter. Let us suppose a collection of experimental data points following a steep (mid-point slope > 0.576) symmetric curve. In principle, the Hill equation would provide a good fit. However, the modified Hill equation could not accommodate the data well, as the only symmetrical curves that it is able to fit are the rectangular hyperbolae (mid-point slope = 0.576). Analysis of the residuals also can be useful in model comparison. Positive and negative residuals should distribute randomly around a horizontal regression line set at 0. A systematic deviation of the residuals from the regression line showing positive and negative clustering in some parts of the plot would be indicative of the inadequacy of the model. This technique could be used to compare non-hierarchical models, for instance, in the above example. Comprehensive revisions of the statistical concepts involved in degree of fit can be found in some excellent monographs and books (Bates & Watts, 1988; Christopoulos & Lew, 2001; Wells, 1992).

3. Mechanistic models Mechanistic models result from applying the physicochemical laws that govern the proposed reaction path to the law of mass action. Mechanistic models differ from empirical ones in the existence of a correspondence between the equation parameters and the chemical constants of the biological process. In general, the use of mechanistic rather than empirical models would be preferable because a deeper insight of the system can be obtained. However, some caution is needed. First, the utility of a mechanistic model will depend on how accurate our understanding of the signal transduction pathways is. This is especially important in GPCRs, for which the transduction of the agonist-receptor interaction into the biological effect is still largely unknown. Second, detailed mechanistic models involving many chemical steps can render mathematical equations inappropriate for non-linear regression analysis because the correlation between parameters and the difficulties in convergence increase with the number of parameters. 3.1. Ionic channels Ionic channels constitute the system par excellence where pharmacological mechanisms can be studied (Colquhoun,

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1998; Lewis et al., 1998). For purposes of illustration, let us now address the issue of the number of receptor-binding sites in the context of the (a)symmetry of E/[A] curves. 3.1.1. A receptor with one binding site The following chemical equilibrium shows the simplest mechanism that can be envisaged for an ionic channel system: a receptor molecule either vacant or complexed with a single agonist molecule in two different states, inactive or active (del Castillo & Katz, 1957). KA

KE

A þ RU ARUAR* where KA = (([A][R])/[AR]) and KE = [AR*]/[AR] (Section A.6.1). Making the substitution x = log[A] in Eq. (A36), the proportion of open channels at equilibrium, p = (bAR * c/ [R0]), is given by p¼

a 1 þ 10xb x

ð6Þ

where a = KE/(1 + KE) and xb = log(KA/(1 + KE)). If we compare Eq. (6) and Eq. (1), we see that the p/x equation for the ionic channel system with one binding site is a symmetric Hill equation with a slope parameter m = 1. The geometric descriptors of the p/x curve are given in Section A.6.1. The fundamental difference with the empirical models described in the previous section is that here the equation parameters can be interpretable in physicochemical terms. They are a function of KA, the agonist-receptor dissociation equilibrium constant and of KE, the equilibrium constant for the opening reaction. 3.1.2. A receptor with n bindings sites either vacant or fully occupied Let us address an extension of the previous case where n sites of the receptor are available for occupation. We will consider an ideal situation in which no receptor molecules are partly occupied. This system can be the result of two unlikely possibilities: (1) n + 1 molecules collide simultaneously or, less outlandishly, (2) the sites are occupied sequentially, with very marked positive cooperativity at each step. The requirement for no intermediate terms is expressed in the following equilibria: KA

for the ionic channel system with n binding sites either empty or fully occupied is a symmetric Hill equation with a slope parameter equal to n, the molecularity of the reaction. In Section A.6.2, the geometric descriptors of the p/x curve are developed. 3.1.3. A receptor with two binding sites where partial occupation is allowed The previous case is not surely found in practice. From a realistic chemical point of view, all the agonist-receptor binding combinations should be considered. The following equation shows the chemical equilibria associated to a receptor with two binding sites. 2kþ1

kþ2

b

k1

2k2

a

2A þ R]A þ AR]A2 R]A2 R* where k + 1 and k + 2 are association rate constants for agonist binding, k  1 and k  2 are dissociation rate constants for agonist unbinding, and a and b are the rate constants for the shutting and opening of the fully occupied channel (Section A.6.3). Making the substitution x = log[A] in Eq. (A48), the proportion of open channels at equilibrium, p = (bA2R * c/ [R0]), is given by p¼

2K

1 þ 1þKAE2 

KE 1þKE KA1 KA2 10x þ 1þK E

 102x

ð8Þ

where KA1 and KA2 are the microscopic equilibrium dissociation constants for the first and second binding processes, respectively, and KE, the equilibrium constant for the opening reaction. The geometric descriptors of the p/x curve are detailed in Section A.6.3. However, it is worthwhile to discuss some properties related to the curve shape. One property commonly discussed is the Hill coefficient. The Hill coefficient at the mid-point, nH50, can be computed as nH50 = 4(dp/ dx)x50/(aln 10), where a = KE/(1 + KE) is the upper

KE

nA þ RU An RUAn R* where KA = [A]n[R]/[AnR] and KE = bAnR * c/[AnR] (Section A.6.2). Making the substitution x = log[A] in Eq. (A42), the proportion of open channels at equilibrium, p = (bAnR * c/ [R0]), is given by p¼

a 1 þ 10nðxb xÞ

ð7Þ

where a = KE/(1 + KE) and xb = (1/n)log(KA/(1 + KE)). If we compare Eq. (7) and Eq. (1), we see that the p/x equation

Fig. 8. Variation of the Hill coefficient at the mid-point, nH50, with the equilibrium constant for the opening reaction (KE) for an ionic channel receptor with two binding sites. KA1 and KA2 are the microscopic equilibrium dissociation constants for the first and the second binding processes, respectively.

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

Fig. 9. Variation of the second derivative of the proportion of active receptors (p) at the mid-point with the equilibrium constant for the opening reaction (KE) for an ionic channel receptor with two binding sites. The curves have been normalized by dividing by the maximum effect (a). KA1 and KA2 are the microscopic equilibrium dissociation constants for the first and the second binding processes, respectively.

asymptote. This expression, which can be obtained directly from Eq. (A5) (see Appendix A), can be a practical way of obtaining the value of nH50 for any E/x equation. Moreover, it can be proven that it is equivalent to the more classical expression nH50 = (d log(E/(a  E))/dx)x50 derived from the linearized form of the Hill equation. Fig. 8 shows the calculated Hill coefficient as a function of log(KE) for three cases: (1) in the absence of cooperativity (KA1 = KA2), (2) in the presence of positive cooperativity (KA1 = 2KA2), and (3) in the presence of negative cooperativity (KA2 = 2KA1). We see that these curves join at n = 2, the molecularity of the process, for high values of KE. However, the curves diverge for low values of KE. The limit values are 1.17 for KA1 = KA2, 1.27 for KA1 = 2KA2, and 1.10 for KA2 = 2KA1. In addition, the Hill coefficient does not depend on the absolute values of the dissociation constants, but on its ratio. To discuss the asymmetry of the p/x curve, we have calculated the second derivative of p relative to x at the midpoint and plotted it against log(KE). Fig. 9 shows that the second derivative tends to 0 (symmetric p/x curve) for high values of KE. The discrepancy in the degree of asymmetry between the p/x curves is greater for low values of KE. A minimum of the second derivatives is found for log(KE) close to 1.

4. A hybrid model: the operational model of agonism Ionic channels constitute a special case in pharmacology because the dependent variable in concentration-effect plots, the proportion of open channels, can be obtained directly from the equilibrium equations. The problem arises when the dependent variable is a functional response, for instance, in the case of GPCRs. Because of the complexity of the signal transduction process, any pharmacological model aiming to include explicitly in the concentration-effect equation the equilibrium constants of the involved chemical

29

reactions inevitably contains a certain degree of arbitrariness; namely, the election of the function linking the product of the reaction mechanism and the observed functional effect. That is why these models could be named hybrid models. They combine the equations from a collection of chemical equilibria (mechanistic models) with an arbitrary transducer function (empirical models). Nevertheless, the efforts that are made to tackle this problem are not in vain if we are aware of the simplifications and limitations of our hypothesis. Because of its wide use, it would be worthwhile to review an example of hybrid models: the operational model of agonism (Black & Leff, 1983). We will start from the equations of the original general case, and we will further discuss more recent applications and developments. 4.1. The general case The operational model of agonism (Black & Leff, 1983) is, from a molecular point of view, a simple model of agonist action. Only the first step of the process, the formation of the agonist-receptor complex, is considered explicitly: KA

A þ RU AR where KA = [A][R]/[AR] (Section A.7). This chemical equilibrium, together with a logistic function for the transduction of receptor occupancy into response, leads to the following E/x function: E¼

Em  tn  10nx ðKA þ 10x Þn þ tn  10nx

ð9Þ

Em is the maximum possible effect in a given tissue, KA is the agonist-receptor dissociation constant, t is the operational efficacy, and n is the slope parameter. The geometric descriptors of the operational model are the upper asymptote, a = Emtn/(1 + tn); the mid-point, x50 = log(KA/((2 + tn)1/n  1)); the mid-point slope, (dE/dx)x = x50 = (0.576Emtnn(2 + tn)((2 + tn)1/n  1))/ ((2 + tn)1/n(1 + tn)2); and, the point of inflection. Full agonists (large value of t) yield a values close to Em, whereas partial agonists (small value of t) yield a values lower than Em. To determine the point of inflection, it is necessary to consider the second derivative of E(x) with respect to x (Section A.7). We note that when n = 1, then d2E/dx2 = 0 at x = x50. Thus, for n = 1, the operational model follows a symmetric curve (Fig. 10a) (Black et al., 1985). This result is consistent because n = 1 Eq. (9) can be rewritten as: E¼

t Em  tþ1 Em  t  10x a ¼ ¼ KA 1 þ 10xb x KA þ 10x þ t  10x tþ1 1 þ 10x

ð10Þ

where a = Em  t/(t + 1) and xb = log(KA/(t + 1)). If we compare Eq. (10) with Eq. (1), we see that the operational model with n = 1 follows a Hill equation with m = 1, that is, the rectangular hyperbola case. Then, the

30

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

curves). In other words, for n = 1, the operational model produces symmetric curves (x50 = xI). For n > 1, the operational model produces asymmetric curves with x50 > xI, and for n < 1, the operational model produces asymmetric curves with x50 < xI. The operational model, like other precedent models (Furchgott, 1966; Stephenson, 1956), lies on an assertion that has been criticized by some authors (Colquhoun, 1998; Jenkinson, 1989). In Eq. (A53), it is established that the total receptor concentration is the sum of the vacant receptor (R) and the agonist-receptor complex (AR). However, it has been found that many conformational states for the ligandreceptor complex (ARi) coexist at equilibrium (Kenakin, 1998; Strange, 1999). Omission of this property is equivalent to considering that [AR] includes all the bounded species of the receptor, and, consequently, the parameter KA of Eq. (A54) does not correspond to a true, but to an apparent, agonist-receptor dissociation constant. This point will be discussed further in Section 5.2. 4.2. Signal transduction by G-protein-coupled receptors: the stoichiometric relationship between the receptor and the G-protein Fig. 10. Concentration-effect (E/x) curves and their first and second derivatives produced by the operational model of agonism. Fixed parameter values for Eq. (9): Em = 1, t = 1, KA = 10  4. a: n = 1. E(x) is symmetrical: the mid-point matches the point of inflection. At x50 =  4.3010, dE/dx is a maximum and d2E/dx2 = 0. b: n = 2. E(x) is not symmetrical: the mid-point (x50 =  3.8645) does not correspond with the point of inflection (xI =  3.9490).

upper asymptote is a and the mid-point and the point of inflection are both equal to xb. When n 6¼ 1, the second derivative equation is rather intractable. However, to prove that the operational model can render an asymmetric curve, we only need to find a system where this property comes true. Let us take, for simplicity, Em = 1, t = 1, and n = 2. Then, (d2E / dx2)x = x50=  0.11 6¼ 0 (Eq. (A62)). Furthermore, this result is independent of the value of KA. Since the mid-point does not correspond with the point of inflection, this theoretical E(x) curve is not symmetrical (Fig. 10b). This particular case corresponds to a partial agonist with a = Em/2. To solve analytically the second derivative equation is an arduous work. Thus, to examine the dependency of the asymmetry of the curve with the operational efficacy and the slope parameters, a graphic approach is taken. Fig. 11 is a contour plot displaying the variation of z = d2E/dx2 at the mid-point with respect to n and t. The n value varies between 0 and 2, whereas t varies between 0 and 20. A value of 0.02 units is chosen for the difference between adjacent isocontour lines. We see that at n = 1, the plot is divided by a vertical line in 2 regions of different sign. For n = 1, then z = 0, irrespective of the value of t (symmetric curves); whereas for n > 1 and n < 1, then z < 0 and z > 0, respectively (asymmetric

The ternary complex model of signal transduction by GPCRs is displayed in the equilibrium: KA

KAR

A þ RU AR þ GU ARG being KA = [A][R]/[AR] and KAR = [AR][G]/[ARG] (Section A.8). Three states of the receptor, vacant (R), agonist-bounded (AR), and the ternary agonist-receptor-G protein (ARG)

Fig. 11. Depiction of (n, t, (d2E/dx2)x50) contour map from operational model with fixed Em = 1. For comparative purposes, the plot has been normalized by dividing the second derivative by the asymptote of the E/[A] curve. The contour lines are given in spaces of 0.02 units. The second derivative of E(x) at the mid-point is 0 (symmetric curve) only for n = 1. For n > 1, (d2E/dx2)x50 is negative, whereas for n < 1, (d2E/dx2)x50 is positive.

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

complex, are included in the mechanism (De Lean et al., 1980). More intricate models for GPCR signal transduction as the extended ternary complex model (Lefkowitz et al., 1993; Samama et al., 1993), the cubic ternary complex model (Weiss et al., 1996a, 1996b, 1996c), the three-state model of agonist action (Leff et al., 1997), or a model with multiple inactive states (Pardo et al., 1997) have been proposed. However, following a parsimony procedure, we aimed to determine the simplest model for GPCRs where asymmetry could be present. There is no reason, apart from simplifying the algebra, for adopting this truncated scheme. However, it is worthwhile to note that, because we have not contemplated the R-G coupling in the absence of ligand, the model cannot account for constitutive activity. From the depicted mechanism, we assume that the ternary complex is the chemical species responsible for the pharmacological response. In a first instance, we will examine the relationship of [ARG] with [A], avoiding the inclusion of an empirical function to yield the observed effect. We will consider two limiting cases, (1) [R0] >> [G0] (Black & Leff, 1983) and (2) [G0] >> [R0] (Mackay, 1988), and the general case, (3) [R0] [G0] (Black & Shankley, 1990; Jenkinson, 1989; Kenakin, 1997a; Pardo et al., 1997). An operational definition of efficacy can be obtained for the first two cases as t = ([R0] / KAR) for [R0] >> [G0] and t = ([G0] / KAR) for [G0] >> [R0]. We see that efficacy is determined by the ratio of the total concentration of either the receptor or the G-protein, which is in excess, and the equilibrium constant for the dissociation of the ternary complex. Including these definitions in Eq. (A72) and Eq. (A78) results in an [ARG]/[A] function formally equivalent to the general E/[A] equation (Eq. (A56)), with n = 1. Consequently, the [ARG]/x function follows symmetric Hill equations when either [R0] or [G0] are in excess. However, when [R0] and [G0] are of the same magnitude, the [ARG]/x function yields an asymmetric curve, as will be shown below. Let us now address with some detail the general case, in which [R0] and [G0] are not neglected, one relative to the other. Fig. 12a depicts the [ARG]/x curves for a constant [G0] = 103 concentration, whereas [R0] ranges from 105 to 102. Following the procedure by Black and Shankley (1990), we can see that log[A50] increases as [R0] decreases from [R0] = 105 until [R0] = [G0] = 103, and that afterwards, log[A50] decreases as [R0] is getting smaller. Interestingly, we have obtained an explicit equation for log[A50] that was not possible to be given in a previous study (Black & Shankley, 1990). Fig. 12b depicts log[A50] as a function of [R0] for a constant [G0] = 103, illustrating the trend observed in Fig. 12a. We can see that log[A50] approaches  7 (the agonist-receptor dissociation apparent constant, see Eq. (A86)) as [R0] ! 0. Eq. (A87) shows the first derivative of [ARG]/x for [R0] = [G0]. However, the second derivative has been omitted because of its length. We have calculated the values of these expressions in a number of situations. For meaningful comparisons, these quantities were divided by the upper asymptote. Remark-

31

Fig. 12. a: Depiction of the concentration of the ternary agonist-receptor-Gprotein complex as a function of the logarithm of agonist concentration (Eq. (A83)). Constant values are KA = 10  4, KAR = 1, and [G0] = 103. [R0] takes the values 105, 104, 3000, 1000, 600, 300, and 100, from left to right. Mid-points are shown as solid circles. b: Logarithm of the mid-point concentration as a function of total receptor concentration, ranging from 0 to 3000. Constant values are KA = 10  4, KAR = 1, and [G0] = 103. It is worthwhile to note that in this figure and throughout the review, [A] stands for free agonist concentration. It is commonly accepted that agonist concentration greatly exceeds the receptor concentration and [A] is approximately equal to total agonist concentration, [A0]. However, if [R0] increases notably, it could cause depletion of the ligand, and then [A] would be different from [A0].

ably, we have observed that the slopes at the mid-point are

0.576 (non-steep curves) and the second derivatives at the mid-point are different from 0 (asymmetric curves). This result, which arises from the mutual depletion of the receptor and the G-protein as the ligand promotes the formation of the complex, is in agreement with what has been found by others (Green et al., 1997; Lee et al., 1986). Because no stoichiometric combinations of [R0] and [G0] can produce steep [ARG]/x curves, an additional property must be included in the mechanism. In ionic channels, midpoint slopes > 0.576 arise from receptor molecules with multiple binding sites. Analogously, a similar hypothesis for steep [ARG]/x curves could be conceived. In GPCRs, the property of several binding sites arises directly when considering dimeric receptors. Receptor dimerization is a rapidly evolving area in pharmacology. Many studies have been presented lately showing that many GPCRs form dimers, among them the b2-adrenergic receptor; the d-opioid receptor; the dopamine D1, D2, and D3 receptors; the

32

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

chemokine CCR2b, CCR4, and CCR5; and the metabotropic glutamate receptor 5 (Bouvier, 2001; Gether, 2000; Milligan, 2001; Tallman, 2000). A quantitative analysis of the relationship between receptor dimerization and curve slope for GPCRs would be worthwhile, but this is beyond the scope of the present review. Nevertheless, it must be said that the idea of cooperative effects among interacting GPCRs is not new. For instance, application of this property provided a mechanistic description of data for cardiac muscarinic receptors (Chidiac et al., 1997) and for D2 dopamine receptors (Armstrong & Strange, 2001). Moreover, detailed analysis of ligand binding by biological macromolecules, thermodynamics, and mathematics can be found in some excellent books (Edsall & Gutfreund, 1983; Wyman & Gill, 1990) and articles (Henis & Levitzki, 1979; Klotz & Hunston, 1979). At this stage, it could be necessary to remark that we have been using [ARG], one of the first species produced in the signal transduction process, as a model of the observed functional effect (E). The steps following the ternary complex formation may determine whether the E profile will resemble that of [ARG] or not. Black and Shankley (1990) have shown that a linear function connecting ARG with E maintains the typical plot depicted in Fig. 12a, in which receptor distribution is detected, whereas a rectangular hyperbolic function masks this occurrence. Then, it could also happen that steep E/[A] curves result from post-[ARG] stimulus-response mechanisms instead of receptor oligomerization.

5. Asymmetry and receptor reserve Receptor reserve is a capital property of pharmacological systems. This term denotes a nonlinear relationship between receptor occupancy and tissue response; that is, maximal responses can be achieved with submaximal receptor occupancy. Moreover, receptor reserve depends on both the tissue and the agonist, its magnitude being directly related to the relative intrinsic efficacies of the agonists (Kenakin, 1997b). A cause-effect relationship between the asymmetry of E/x curves and the existence of receptor reserve has been proposed recently (Agneter et al., 1997; Feuerstein & Limberger, 1999). In our opinion, this assessment, together with the suggestion that there is also a relationship between Hill coefficients > 1 and receptor reserve (Agneter et al., 1997; Hey et al., 1994), need further study and discussion. Although this issue was the subject of a recent debate (Agneter et al., 1998; Giraldo, 1998), a detailed analysis using published experimental data from our laboratory (Tabernero et al., 1999; Vivas et al., 1997) and the foregoing equations could provide a better understanding of the problem. Two biological systems, the vas deferens and endothelium-denuded aortic rings from 16- to 18-weekold male Wystar-Kyoto (WKY) rats and spontaneously

hypertensive rats (SHR), will be examined from the viewpoint of symmetry. 5.1. The pharmacological systems 5.1.1. System 1. Contractile response to norepinephrine in a1-adrenoceptor vas deferens from spontaneously hypertensive and Wystar-Kyoto rats The experiments were performed in segments (1 cm in length) of the epididymal half of the rat vas deferens, as described in Vivas et al. (1997). Non-cumulative norepinephrine-induced increases in isometric tension were registered with 5 min intervals between concentrations. To characterize agonist function, partial receptor inactivation is required. Tissues were exposed for 10 min to the alkylating agent phenoxybenzamine (0.1 mM), and a second noncumulative curve was performed in the same conditions. 5.1.2. System 2. Contractile response to phenylephrine in endothelium-denuded aorta rings from spontaneously hypertensive and Wystar-Kyoto rats The thoracic aorta was cleaned of fat and connective tissue and cut into segments 5 mm long. To obtain endotheliumdenuded rings, the intimal surface was carefully rubbed with a wooden stick, as described in Tabernero et al. (1999). Each ring was contracted with norepinephrine (0.01 mM) and relaxed with acetylcholine (1 mM). Preparations were considered denuded when they failed (0%) to relax in response to acetylcholine. Cumulative E/[A] curves to phenylephrine (3 nM –300 mM) were constructed. Afterwards, tissues were exposed for 10 min to the alkylating agent phenoxybenzamine (10 nM) to allow partial a1-adrenoceptor inactivation, and the agonist E/x curve was repeated. 5.2. Pharmacological analysis: the operational model of agonism 5.2.1. Application to System 1 Table 1 shows the parameter estimates of the operational model of agonism (Em, n, log t, pKA) and the resulting geometric properties (a, x50, slope at the mid-point, and the second derivative of the response at the mid-point) of E/x curves for the contractile response of rat vas deferens to norepinephrine (Vivas et al., 1997) (see Section 8 for data analysis). We observe that SHR produce a symmetric curve because the second derivative of the response at the midpoint is not significantly different from 0 (the 95% confidence interval (CI) includes 0). This result is consistent with an n value not significantly different from 1. Interestingly, curves for WKY rats show a different behavior. The E/x curve is asymmetric because the second derivative of the response at the mid-point is significantly different from 0 (the 95% CI does not include 0). This result is consistent with a value of n significantly different from 1. However, contrary to what has been postulated (Agneter et al., 1997), no relationship between receptor reserve and asymmetry is

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45 Table 1 Parameter estimates and geometric descriptors for norepinephrine-induced contraction from WKY rats and SHR vas deferens by the operational model of agonism SHR

WKY

Mean

95% CI

Mean

95% CI

25.52 4.89 1.09 1.81 24.99  6.71

(23.62, 27.43) (4.63, 5.16) (0.94, 1.23) (1.54, 2.08) (23.04, 26.94) (  6.84,  6.59)

15.99 4.89 1.62 1.67 15.87  6.55

(13.81, 18.16) (4.52, 5.26) (1.17, 2.06) (1.39, 1.95) (13.74, 17.99) (  6.66,  6.43)

0.62  0.17

(0.54, 0.70) (  0.40, 0.05)

0.90  0.75

(0.67, 1.13) (  1.32,  0.17)

Em pKA n Log t a ¼ limx!1 E x50 = x for E = a/2 (dE/dx)x = x50 (d2E/dx2)x = x50

Mean and 95% CI values were obtained for each strain by fitting each pair of E/x curves for norepinephrine (control and after exposure to phenoxybenzamine) to Eq. (9). Thirteen and 12 pair of curves were used for SHR and WKY rats (Vivas et al., 1997). For clarity, only values for the control curves are shown.

observed. The operational model of agonism proposes Eq. (11) as the mathematical function for the transduction of receptor occupancy into a response. It should be noted that this equation is the same as Eq. (9) when binding proceeds according to the scheme A + R U AR with KA = [A][R]/[AR].
n Em ½AR ½R 
0 n E¼ ð11Þ ½AR 1 tn þ ½R 

33

In spite of these drawbacks, we believe that Eq. (11) could still be useful for estimation of receptor occupancy if we do not pretend to dissect the equilibrium dissociation constants of all of the receptor-bounded species. Bearing this premise in mind, [AR] would now represent the total concentration of bounded receptor, whereas the dissociation constant parameter of Eq. (A54) would not be a true, but an apparent, dissociation constant. Let us describe this idea in more detail. The total receptor concentration can be expressed as: ½R0  ¼ ½R þ ½AR1  þ ½AR2  þ : : : þ ½ARn 

ð12Þ

where R is the free receptor and AR1, . . ., ARn are the agonist-bounded receptor species, at least one of them being an active state of the receptor (Pardo et al., 1997). Each [ARi] is determined by a collection of equilibrium constants, KAi, whose values are precluded in this context. We denote receptor occupation, [AR], by: ½AR ¼ ½AR1  þ ½AR2  þ : : : þ ½ARn 

ð13Þ

where AR is not a physical entity, but the ensemble of occupied receptors, ARi, which are spontaneously interconvertible on the time scale of the experiment. An apparent agonist-receptor dissociation constant is defined as: KA ¼

½A  ½R ½AR

ð14Þ

0

Eq. (11) is an empirical function and, therefore, is not supported by any chemical mechanism. It simply states that transduction of occupancy into a response depends on operational efficacy (t) and that full agonists (t high) do not need to occupy all the receptors to yield the maximum possible response, Em. In this context, inclusion of the parameter n is again empirical, although necessary to account for E/[A] curves that are different from a rectangular hyperbola. In Eq. (11), receptor occupancy is represented by [AR], a single agonist-receptor species. The possibility of an activated state of the receptor (AR*) or the binding of the agonist-receptor pair to a transducer protein (G-protein) to form a ternary ARG complex is not included. Thus, it is implicitly assumed that the AR-E relationship does not disturb the A-AR relationship. This is a simplification that, as it has been pointed out, would lead to wrong estimates of agonist-receptor dissociation constants (Colquhoun, 1987, 1998). This problem has been reconsidered within the framework of the operational model by including the concentration of the ARG complex as a component of the total receptor concentration (Eqs. (A81 – 86)) (Black & Shankley, 1990). However, an unavoidable degree of uncertainty exits in the proposal of a particular function for transformation of the ARG concentration into a response, obtaining different E/x curves if we use, for instance, a linear or a logistic transducer function (Black, 1996; Black & Shankley, 1990).

We see that because [AR] contains the population of inactive and active states of bounded receptor; KA includes implicitly a component of agonist efficacy. Following the scheme of the operational model of agonism, a logistic function can be proposed for the transduction of occupation into a response. E¼

Em  ½ARn KnE þ ½ARn

ð15Þ

By dividing the numerator and the denominator of the second member of Eq. (15) by [R0]n and then redefining [R0]/KE as t, Eq. (11) is obtained. Fig. 13 depicts the theoretical E/x (Fig. 13a solid lines; Eq. (9)) and E/([AR]/[R0]) (Fig. 13b; Eq. (11)) curves for SHR and WKY rats using the operational parameters Em, n, t, and KA from Table 1. For comparative purposes, the effects in Fig. 13b are divided by the maximum responses, a = Emtn/(1 + tn). Fig. 13b clearly shows the existence of a large receptor reserve for both systems: < 25% of receptor occupancy is necessary to yield the maximum response. Interestingly, at the point ([AR]/[R0] = 0.044, E/a = 0.765), the curves cross. For [AR]/[R0] < 0.044, the receptor reserve is greater for SHR, whereas for [AR]/[R0] > 0.044, the receptor reserve is greater for WKY rats. This is a consequence of the difference in the values of t and n parameters between strains: t is higher for SHR, whereas n is higher for WKY rats.

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J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45 Table 2 Parameter estimates and geometric descriptors for phenylephrine-induced contraction from WKY rats and SHR endothelium-denuded aortic rings by the operational model of agonism SHR

Em pKA n Log t a ¼ limx!1 E x50 = x for E = a/2 (dE/dx)x = x50 (d2E/dx2)x = x50

WKY

Mean

95% CI

Mean

95% CI

15.23 6.39 1.12 0.77 12.68  7.21

(8.40, 22.07) (6.17, 6.61) (0.86, 1.38) (0.41, 1.13) (6.81, 18.55) (  7.41,  7.01)

20.43 6.27 1.01 1.36 18.74  7.67

(17.31, 23.55) (6.13, 6.40) (0.89, 1.13) (1.13, 1.58) (16.46, 21.02) (  7.78,  7.57)

0.61  0.19

(0.51, 0.70) (  0.64, 0.25)

0.59 0.05

(0.52, 0.65) (  0.22, 0.32)

Mean and 95% CI values were obtained for each strain by fitting each pair of E/x curves for phenylephrine (control and after exposure to phenoxybenzamine) to Eq. (9). Eight and 16 pair of curves were used for SHR and WKY rats (Tabernero et al., 1999). For clarity, only values for the control curves are shown.

Fig. 13. a: Concentration-effect curves for norepinephrine-induced contraction in the epididymal portion of vas deferens from WKY rats (., n = 12) and SHR (&, n = 13) using Eq. (9) and parameter values (Em, KA, t, and n) from Table 1 (solid lines) and using Eq. (4) and parameter values (a, xb, and p) from Table 4 (broken lines). For clarity, in the operational model, only control curves and not those after exposure to the alkylating agent are shown. b: Relative receptor occupancy-effect curves from WKY rats and SHR vas deferens using Eq. (11) and parameter values (Em, t, and n) from Table 1. For purposes of comparison, calculated effects of each curve are divided by their maximum response (a). exp. values, experimental mean values ± SEM.

These results allow us to conclude that receptor reserve does not induce asymmetry into E/x curves. Both strains present a high and similar receptor reserve, but the resulting E/x curves are qualitatively different relative to the property of symmetry: SHR display a symmetric curve, whereas WKY rats do not. 5.2.2. Application to System 2 Table 2 shows the parameter estimates of the operational model of agonism (Em, n, log t, pKA) and the resulting geometric properties (a, x50, slope at the mid-point, and the second derivative of the response at the mid-point) of E/x curves for the phenylephrine-induced contraction of aorta denuded rings (Tabernero et al., 1999) (see Section 8 for data analysis). Here, both SHR and WKY rats produce symmetric curves because the second derivatives of the responses at the mid-point are not significantly different from 0 (the 95% CI includes 0). These results are consistent with an n value not significantly different from 1. We can also see that WKY rats present a higher value of the

Fig. 14. a: Concentration-effect curves for phenylephrine-induced contraction in endothelium-denuded aortic rings from WKY rats (., n = 16) and SHR (&, n = 8) using Eq. (9) and parameter values (Em, KA, t, and n) from Table 2 (solid lines) and using Eq. (4) and parameter values (a, xb, and p) from Table 5 (broken lines). For clarity, in the operational model, only control curves and not those after exposure to the alkylating agent are shown. b: Relative receptor occupancy-effect curves from WKY rats and SHR using Eq. (11) and parameter values (Em, t, and n) from Table 2. For purposes of comparison, calculated effects of each curve are divided by their maximum responses (a). exp. values, experimental mean values ± SEM.

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

35

operational efficacy (t) than SHR. Thus, we will expect a larger receptor reserve for WKY rats. Fig. 14 depicts the theoretical E/x (Fig. 14a solid lines; Eq. (9)) and E/([AR]/ [R0]) (Fig. 14b; Eq. (11)) curves for SHR and WKY rats using the operational parameters Em, n, t, and KA from Table 2. It can be seen (Fig. 14b) that a remarkable receptor reserve is present in WKY rats, whereas this property is less significant in SHR. Analogously to System 1, it can be concluded that there is no relationship between receptor reserve and the symmetry of E/x curves. The great difference found in receptor reserve between strains does not yield a difference in the symmetry of the curves. 5.3. Pharmacological analysis: the logistic fitting procedures We have shown that the operational model of agonism can properly fit asymmetric E/x experimental data. We will now turn our attention to the proposed extensions of the Hill equation. The Richards function failed to give reliable parameter estimates when it was fitted to our data. Either very large intervals of confidence were obtained or, simply, it did not converge. The Gompertz model converged in all the cases and yielded acceptable standard errors for the parameter estimates. However, the Gompertz model invariably produces asymmetric curves, and, thus, it is not a suitable mathematical function when the experimental data are symmetric. The modified Hill equation is potentially able to model the transition from symmetric to asymmetric curves. Regretfully, although the modified Hill equation has the same number of parameters than the Hill equation or the Gompertz model, in our experience, it is less efficient for data fitting. We found in a number of cases a high correlation between the parameters xb and p of the modified Hill equation (R > 0.9 in absolute value). This strong correlation Table 3 Parameter estimates and geometric descriptors for an individual SHR vas deferens E/[A] curve fitted by empirical models

Hill equation

Richards function

Gompertz model Modified Hill equation

a xb m a xb m s a xI m a xb p

Parameter estimates

95% CI

Residual sum of squares df

22.06 6.47 1.10 22.73 6.88 0.87 1.96 23.98 6.67 0.63 22.26 6.63 1.32

(21.01, 23.11) (  6.54,  6.39) (0.94, 1.26) (21.11, 24.35) (  7.82,  5.94) (0.53, 1.21) (  0.71, 4.62) (21.85, 26.10) (  6.76,  6.57) (0.50, 0.77) (21.59, 22.94) (  6.83,  6.42) (0.91, 1.72)

0.4561 4

± ± ± ± ± ± ± ± ± ± ± ± ±

0.38 0.03 0.06 0.51 0.30 0.11 0.84 0.76 0.03 0.05 0.24 0.07 0.15

0.1952 3

Fig. 15. Experimental data points and fitted curves with the four empirical models (see Table 3).

was not observed in our data between the analogous parameters xb and m of the Hill equation, and xI and m of the Gompertz model. To illustrate the differences in curve fitting among these logistic models, let us take the values of an individual concentration-effect curve from System 1 as an example (see Table 3). Nested models can be compared by the extra sum of squares analysis (see Section 2.5). Substituting the corresponding residual sums of squares and degrees of freedom in Eq. (5), the F values for the comparison between the Richards model and the Hill equation, the Gompertz model, and the modified Hill equation were 4.01, 6.83, and 1.28, respectively, not statistically significant at the 5% level of confidence. As the Richards model does not improve the fitting by the Hill equation, we can infer that the curve is symmetric (Richards s parameter is equal to 1). Moreover, because the Richards model does not improve the fitting by the modified Hill equation, we can assume that the curve is a rectangular hyperbola (Richards s and m parameters both equal to 1). Fig. 15 shows the theoretical curves produced for all 4 models superimposed on the data. In addition, standard errors and confidence intervals of the parameter estimates are shown in Table 3. Standard errors and confidence intervals yielded by direct nonlinear regression fitting to a single curve are approximate and cannot be

Table 4 Parameter estimates and geometric descriptors for norepinephrine-induced contraction from WKY rats and SHR vas deferens by the modified Hill equation SHR

0.6397 4 0.2787 4

Parameter estimates result from applying empirical models developed in the text to the following raw data expressed as ([norepinephrine] mM, effect mN): (0.01, 0.00), (0.03, 1.23), (0.1, 4.66), (0.3, 10.29), (1, 16.91), (3, 19.85), (10, 21.81).

a xb p x50 (dE/dx)x = x50 xI

WKY

Mean

95% CI

Mean

95% CI

25.40  6.68 1.09  6.69 0.57  6.68

(22.43, 28.37) (  6.86,  6.50) (0.68, 1.50) (  6.90,  6.48) (0.51, 0.63) (  6.86,  6.50)

16.04 6.74 1.72  6.45 0.65  6.51

(13.44, 18.64) (  6.96,  6.51) (1.39, 2.05) (  6.66,  6.24) (0.63, 0.68) (  6.72,  6.30)

Mean and 95% CI values were obtained for each strain by fitting each E/x curve to Eq. (4). Nine and 8 curves whose model fittings converged satisfactorily for SHR and WKY rats, respectively, were used.

36

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

Table 5 Parameter estimates and geometric descriptors for phenylephrine-induced contraction from WKY rats and SHR endothelium-denuded aortic rings by the modified Hill equation SHR

a xb p x50 (dE/dx)x = x50 xI

WKY

Mean

95% CI

Mean

95% CI

12.84  7.26 1.09  7.25 0.58  7.24

(7.96, 17.72) (  7.52,  7.00) (0.81, 1.38) (  7.42,  7.07) (0.53, 0.62) (  7.44,  7.05)

19.42  7.60 1.09  7.61 0.57  7.60

(16.39, 22.45) (  7.82,  7.39) (0.83, 1.35) (  7.72,  7.50) (0.53, 0.62) (  7.73,  7.46)

Mean and 95% CI values were obtained for each strain by fitting each E/x curve to Eq. (4). Nine and 13 curves whose model fittings converged satisfactorily for SHR and WKY rats, respectively, were used.

used to obtain probabilistic conclusions. However, they can give us some indication of the reliability of parameter estimates. We see that the standard errors produced by the Richards function are comparatively very great. The 95% CIs of xb (responsible for curve location) and s (responsible for asymmetry) are too large, indicating that xb and s are not to be used for estimation. The modified Hill equation yielded smaller standard errors, and, consequently, the CIs are more reasonable (see Section 8 for a description about data analysis and error estimation). This property repeated in all the fittings we performed. In addition, the Richards model failed to converge any more often than the modified Hill equation. Accordingly, we decided to choose the modified Hill equation as the empirical model to compare with the operational model of agonism on the analysis of symmetry. In our systems, an agreement between the modified Hill equation and the operational model in assessing the properties of symmetry of E/x curves was found (compare Table 4 with Table 1 and Table 5 with Table 2). Table 4 shows that the theoretical curve of the contractile response to norepinephrine in a1-adrenoceptor vas deferens from SHR is symmetric (the 95% CI for the mean of the slope parameter p includes the value 1), whereas that from WKY rats is asymmetric (the 95% CI for the mean of the slope parameter

p does not include the value 1). In addition, Table 5 shows that the theoretical curves of the contractile response to phenylephrine in endothelium-denuded aortic rings from SHR and WKY rats are both symmetric (the 95% CI for the mean of the slope parameter p includes the value 1). Figs. 13a and 14a (broken lines) also show the curves produced by the modified Hill equation when the parameters from Eq. (4) are substituted by the corresponding values from Tables 4 and 5. It can be seen that our data are fitted similarly by the operational and the modified Hill models.

6. Are the Hill coefficient and receptor reserve two related properties? The question remains: If there is no relationship between the shape of the E/x curve and the existence of receptor reserve, what experimental situation led to the proposal that a receptor reserve induces a slope parameter (m) > 1 of a curve fitted with the Hill equation? (Agneter et al., 1997). As we will show from the following Scheme 1, this is simply a particular case. Scheme 1 shows the mid-point slopes obtained from the Hill equation and from the operational model of agonism (Black et al., 1985). To allow meaningful comparisons of curves with different asymptotes, the values have been normalized by dividing the resulting derivatives by the asymptotes (a) of the E/x curves. If we assume that the operational model and the Hill equation produce a common value for the mid-point slope, the following relationship using Eqs. (i) and (ii) from Scheme 1 is obtained (Black et al., 1985). m¼

n  ð2 þ tn Þ  ðð2 þ tn Þ1=n  1Þ ð2 þ tn Þ1=n  ð1 þ tn Þ

ð16Þ

Eq. (16) is a general expression, and different particular cases can be considered. Case (a): there is a substantial receptor reserve (t >> 1). Then, m n. Because n is independent of t (see Eq. (9)), we can conceive systems with n greater, equal, or lower than 1. Accordingly, we can

Scheme 1. Mid-point slopes of normalized E/x curves by the Hill equation and the operational model of agonism.

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

Fig. 16. Simulation of the slope parameter m (Hill equation) versus the slope parameter n (operational model of agonism), assuming that the operational efficacy t << 1 (no receptor reserve) and common values of the mid-point slopes for both the Hill and the operational approaches.

find systems with receptor reserve presenting values of m greater, equal, or lower than 1. It can be concluded that systems with receptor reserve do not necessarily produce steep (m > 1) curves when modeled with the Hill equation. Case (b): there is not receptor reserve. Let us suppose the limiting case in which t << 1. Then (Black et al., 1985), Eq. (16) leads to:   1 m ¼ 2n  1  1=n ; if   1 ð17Þ 2 Fig. 16 shows the plot of Eq. (17). Once more, we can consider systems with n greater, equal ,or lower than 1. The Hill coefficient m varies from values lower than 1 to values below the limiting value of 2ln2 (  1.39). This upper limit increases for t values not negligible relative to 1. For instance, substituting in Eq. (16) t equal to 5 (which is not greater enough to cause receptor reserve) and n equal to 4 produces a value of m equal to 3.21 > 1. From the foregoing application of the operational model of agonism, we see that, apparently, receptor reserve and the Hill coefficient are two independent properties. Presence and absence of receptor reserve theoretically can be possible with Hill coefficients either greater, equal, or lower than 1.

7. Concluding remarks A number of empirical and mechanistic models have been examined for their ability to assess the shape of concentration-effect curves. The operational model (Eq. (9)) does not predict a relationship between receptor reserve and the asymmetry of E/x curves. This relationship has not been found in the examples we have presented. However, this cannot preclude that such a relationship could emerge in some other situation. Similarly, a Hill coefficient > 1 was not indicative of the existence of receptor reserve. Although there is no reason to expect one on theoretical grounds, the possibility that a relationship could appear in some other systems or circumstances cannot be excluded. Also, the Hill

37

equation, the Richards function, and the Gompertz model were unable to distinguish between experimental symmetric or asymmetric E/x curves either inherently, because of the equation definition (the Hill equation and the Gompertz model are functions intrinsically symmetric and asymmetric, respectively), or technically, because of pitfalls during the nonlinear regression fits (the Richards function). Yet, the modified Hill equation and the operational model of agonism were instrumental to this purpose. In both cases, a parameter value (p in the case of the modified Hill equation and n in the case of the operational model) different from 1 was the condition for the existence of asymmetry. In the operational model of agonism, the n parameter is introduced as a component of the transducer function (Eq. (11)). Consequently, it would be expected that the property of asymmetry be related to the molecular mechanisms responsible for signal transduction. Applied mechanistic models have shown that in the case of ionic channels, the asymmetry of the curves is related to the case of receptor molecules with multiple binding sites, where partial occupation of the receptor molecule is allowed. In the case of GPCRs, the asymmetry is related to receptor distribution, in which [R0] and [G0] are not negligible, one relative to the other. Moreover, an explicit equation was obtained for the mid-point of the [ARG]/x function that allowed the calculation of the slope at the mid-point. Interestingly, no steep curves (slope > 0.576) were obtained. To explain the steepness of experimental concentration-effect curves for GPCRs without making use of empirical functions, a proposal about a relationship between steep concentration-effect curves and receptor oligomerization was suggested. In this regard, it should be emphasized that new efforts pursuing the integration in model equations of data coming from complementary fields, amongst them pharmacology, molecular biology, and computational chemistry, would be of great help for a better understanding of physiological function.

8. Computational details, data analysis, and software Derivatives of concentration-effect equations were obtained by using Mathematica (Wolfram Research Inc., Champaign, IL, USA), a program of symbolic calculus. Curves were plotted with SigmaPlot 2000 (SPSS Inc., Chicago, IL, USA). Statistical analyses were carried out with SAS/STAT1 (SAS Institute Inc., Cary, NC, USA) statistical package. PROC NLIN was performed for nonlinear regression analysis. Marquardt method was used to fit the models to the data. No weighting scheme was applied. A set of reasonable starting values, instead of only one value, was given to each parameter. This generates a grid, and the program calculates the residual sums of squares for each point of the grid (each combination of possible starting values). Although this can take a long time, it ensures a good choice of starting values for the iterative least squares process.

38

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

In Tables 1 and 2, the operational model of agonism was used for parameter estimation. The multiple (two) curve design and the inactivation method was applied (Leff et al., 1990). For each strain (SHR and WKY), each pair of experimental curves (control and after exposure to phenoxybenzamine) was fitted with the operational model, providing a common estimate of Em, n, and pKa and a value of log t for each curve. Normally, one concentration of the alkylating agent is sufficient to compute t. Nevertheless, to test for internal consistency, we performed preliminary experiments at different concentrations of phenoxybenzamine. For simplicity, only the value of t for the control curve is shown. From the sample of estimates, mean values and standard errors were calculated for each parameter. CIs were produced using the Student’s t distribution. See Vivas et al. (1997) and Tabernero et al. (1999) for detailed descriptions of the experiments. In Tables 4 and 5, the modified Hill Eq. (A27) was used to fit each individual control curve for both strains. Mean values, standard errors, and CIs were calculated analogously to the operational model. The curves represented in Figs. 13a and 14a are the theoretical control curves obtained by substituting the mean values of the parameters from Tables 1 and 4 (Fig. 13) and 2 and 5 (Fig. 14) in the corresponding equations. To let the readers check the fitting procedures, one single experimental curve, whose data are explicitly shown, was analyzed in Table 3 by all the empirical models presented in this review. It is worth noting that standard errors and CIs obtained in this way are approximate. They have been displayed to stress that caution should be taken with the values of parameter estimates when the standard errors produced directly by the fittings are very large.

We assume that an increase of the [A] leads to an increase of the observed effect and, accordingly, that the slope parameter m is > 0. The upper asymptote, a, is the value towards the effect approaches as [A], and, consequently, x, increases. It is obtained by taking the limit of E as x goes to + 1. a ¼ lim E ¼ a: x!1

ðA2Þ

The mid-point, x50, is the value of x that yields E = a/2. Substituting E = a/2 in Eq. (A1) and solving the equation for x, we find: x50 ¼ xb

ðA3Þ

The mid-point slope, (dE/dx)x = x50, is the value of the slope at the mid-point. Taking the first derivative of the effect with respect to x and substituting x with x50 gives: dE 10mðxb xÞ  a  m  ln10 ¼ dx ð1 þ 10mðxb xÞ Þ2 

dE dx

 ¼ x¼x50

a  m  ln10 ¼ 0:576  a  m 4

ðA4Þ

ðA5Þ

The point of inflection, xI, is a point on the curve at which the first derivative is a maximum or minimum and the second derivative is 0. In our case, because m is positive, dE/dx is a maximum at x = xI. To determine the point of inflection, it is straightforward to find the value of x such that makes the second derivative equal to 0.

Acknowledgments This work was supported in part by CICYT (SAF990073; PM98-0178), DGES (PB98-0907), Fundacio´ La Marato´ de TV3 (1014/97), and Grups de Qualitat de Recerca (1999SGR-00119). J.G. and E.V. are members of the EC Vascan 2000 Consortium (QLG1-1999-00084). Special thanks are given to the anonymous referees of this paper for their criticisms and suggestions.

A.1. The Hill equation The E/x function is derived from the E/[A] relationship by defining x = log[A] and xb = log b. a  ½Am a a   ¼ ¼ b þ ½Am 1 þ 10bx m 1 þ 10mðxb xÞ m

ðA6Þ

By setting the second derivative equal to 0, we see that it is necessary that the term (  1 + 10m(xb  xIÞ) be 0, and, consequently, the root is: xI ¼ xb

ðA7Þ

A.2. The Richards function

Appendix A. The E/x functions and their geometric descriptors

E¼

d2 E 10mðxb xÞ  ð1 þ 10mðxb xÞ Þ  a  m2 ln100 ¼ dx2 ð þ10mðxb xÞ Þ3

ðA1Þ

To assess the geometric descriptors of the Richards function, we will proceed analogously to the Hill equation fitting. The E/x function is derived from the E/[A] relationship by defining x = log[A] and xb = log b. E¼

a  ½Ams a a ¼  b m s ¼ mðx ðb þ ½Am Þs ð1 þ 10 b xÞ Þs 1 þ 10x m

ðA8Þ

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

The upper asymptote is given by a ¼ lim E ¼ a:

The upper asymptote is found to be ðA9Þ

x!1

The mid-point, x50, is obtained by putting E = a/2 in Eq. (A8) and solving the equation for x. 1 logð21=s 1Þ m

ðA10Þ

To find the mid-point slope, we derive E with respect to x and substitute x with x50 in the resulting relationship. dE 10mðxb xÞ a  m  s  ln10 ¼ dx ð1 þ 10mðxb xÞ Þsþ1 

dE dx

 ¼


 1 a  m  ln10  s 1  21=s

x¼x50

2

ðA11Þ

ðA12Þ

To determine the point of inflection, we derive Eq. (A11) with respect to x and set the resulting expression equal to 0. d2 E 10mðxb xÞ ð1 þ s  10mðxb xÞ Þ  a  m2 s  ln100 ¼ dx2 ð1 þ 10mðxb xÞ Þsþ2 ðA13Þ In order for Eq. (A13) to be 0, it is required that the term (  1 + s10m(xb  x)) be 0, and, therefore, the root is given by: xI ¼ xb þ

a ¼ lim E ¼ a x!1

x50 ¼ xb 

1 logs m

ðA14Þ

We see that to yield curves with a point of inflection, s must be > 0. Finally, by substituting Eq. (A14) in Eq. (A8) we obtain the E at the point of inflection. a s Eðx ¼ xI Þ ¼  1 þ 1s

39

ðA15Þ

Note that the symbols n or d with the property n = d = 1/s can be found in the literature instead of s.

ðA17Þ

The mid-point, x50, is obtained by putting E = a/2 in Eq. (A16) and solving the equation for x. x50 ¼ xI 

1 logðln2Þ m

ðA18Þ

The mid-point slope can be determined by taking the derivative of E with respect to x and letting x = x50 in the resulting relationship. dE 10mðxI xÞ  a  m  ln10 ¼ mðx xÞ dx e10 I 

dE dx

 ¼ x¼x50

a  m  ln10  ln2 ¼ 0:798  a  m 2

ðA19Þ

ðA20Þ

To find the point of inflection, we derive Eq. (A19) with respect to x and set the resulting expression equal to 0. d2 E 10mðxI xÞ ð1 þ 10mðxI xÞ Þ  a  m2 ln100 ¼ mðx xÞ dx2 e10 I

ðA21Þ

For Eq. (A21) be equal to 0, it is necessary that the term (  1 + 10m(xI  x)) be 0, and, consequently, the point of inflection is the parameter xI. The E at the point of inflection is obtained by substituting xI into Eq. (A16). a a Eðx ¼ xI Þ ¼ < Eðx ¼ x50 Þ ¼ e 2

ðA22Þ

A.4. The Gompertz model and the Richards function It can be proven that the Richards function tends to the Gompertz model as the asymmetric parameter s of the Richards function increases. The Richards function was written above as:

A.3. The Gompertz model E¼ The Gompertz model (Gompertz, 1825) may be defined

a ð1 þ

10mðxb xÞ Þs

ðA8Þ

as: E¼

a e

10mðxI xÞ

where x = log[A].

but, ðA16Þ xI ¼ xb þ

1 logs m

ðA14Þ

40

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

Then, by substituting xb from Eq. (A14) into Eq. (A8), we find E¼


a mðxI xÞ

1 þ 10

s

 s Making use of the property lims!1 1 þ zs ¼ ez, we see that the Gompertz model is the limiting case of the Richards function, as s ! 1. s!1

s!1

a 1þ

s ¼ mðx xÞ

10

I

s

a mðxI xÞ

e10

ðA24Þ

If the Richards function tends to the Gompertz model as s increases, the same will happen with the geometric descriptors. For instance, we see that the degree of asymmetry, measured as the difference between xI and x50, from the Richards function tends to the value given by the Gompertz model when s ! 1.   1 logðs  ð21=s 1ÞÞ lim ðxI x50 Þ ¼ lim s!1 s!1 m ¼

dE dx

ðA23Þ

s

lim E ¼ lim




1 logðln2Þ m

ðA25Þ

Similarly, for the effect at the point of inflection: a a s ¼ lim Eðx ¼ xI Þ ¼ lim  1 s!1 s!1 1 þ e s

ðA26Þ

The E/x function is obtained from the E/[A] relationship by defining x = log[A] and xb = log b. ðA27Þ

The upper asymptote is given by a ¼ lim E ¼ a x!1

ðA28Þ

The mid-point, x50, is assessed by putting E = a/2 in Eq. (A27) and solving the equation for x. x50 ¼ xb logð21=p 1Þ

ðA29Þ

To find the mid-point slope, we derive E with respect to x and substitute x with x50 in the resulting relationship. dE 10xb x a  p  ln10 ¼ dx ð1 þ 10xb x Þpþ1

¼


 1 a  ln10  p 1  21=p

ðA31Þ

2

x¼x50

To determine the point of inflection, we derive the Eq. (A30) with respect to x and set the resulting expression equal to 0. d2 E 10xb ð10x þ p  10xb Þ  a  p  ln100 ¼ dx2 ð10x þ 10xb Þ2 ð1 þ 10xb x Þp

ðA32Þ

To make Eq. (A32) equal to 0, it is necessary that the term (  10x + p10xb) be 0, and, consequently, the point of inflection is: xI ¼ xb þ logp

ðA33Þ

By substituting Eq. (A33) into Eq. (A27), we have the effect at the point of inflection. Eðx ¼ xI Þ ¼


a 1 þ 1p

p

ðA34Þ

A.6. Ionic channels

A.5. The modified Hill equation

a  ½Ap a a p ¼ E¼ p ¼  b ð1 þ 10xb x Þp ðb þ ½AÞ 1 þ 10x



ðA30Þ

A.6.1. A receptor with one binding site The simplest mechanism for an ionic channel is the del Castillo and Katz mechanism. A single agonist molecule binds the receptor in a shut state, which afterwards isomerizes to an open state (del Castillo & Katz, 1957). KA

KE

A þ RU ARUAR*

ðA35Þ

where KA is the agonist-receptor dissociation constant, KA = ([A][R]/[AR]); AR and AR* are the shut state and the open state, respectively; and KE is the equilibrium constant for the opening reaction, KE = [AR*]/[AR]. The corresponding property in ionic channel systems to the observed effect in agonist-receptor systems is the proportion of open channels at equilibrium. p¼

½AR* KE ½A ¼ ¼ ½R0  KA þ ð1 þ KE Þ  ½A

KE 1þKE ½A KA 1þKE þ ½A

ðA36Þ

Making the substitution of x = log[A] and comparing Eq. (A36) with Eq. (A1), we see that p/x gives a Hill (symmetric) equation with the slope parameter m = 1. The geometric parameters are found easily by simple substitution. The upper asymptote is: a ¼ lim p ¼ x!1

KE 1 þ KE

ðA37Þ

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

The mid-point is: KA x50 ¼ log 1 þ KE 

The mid-point slope is:  dp KE ln10 ¼ dx x¼x50 4  ð1 þ KE Þ

ðA38Þ

ðA39Þ

ðA40Þ

A.6.2. A receptor with n bindings sites either vacant or fully occupied Let us suppose a receptor with n binding sites and an ideal situation: all the receptor sites are vacant or fully occupied, either in a shut state or in an open state. KA

KE

nA þ RU An RUAn R*

A.6.3. A receptor with two binding sites where partial occupation is allowed Let us derive the equations to a more realistic case in which two binding sites can be total or partially occupied.

ðA41Þ

where KA = [A]n[R]/[AnR], and KE = bAnR * c/[AnR] The proportion of open channels at equilibrium is: ½A R* KE ½An p¼ n ¼ ½R0  KA þ ð1 þ KE Þ  ½An ¼

kþ2

b

k1

2k2

a

ðA42Þ

¼

bA2 R *c ½R0  KE ½A2 KA1 KA2 þ 2  KA2 ½A þ ð1 þ KE Þ  ½A2

x!1

a ¼ lim p ¼ x!1

KE 1 þ KE

ðA43Þ

The mid-point is: x50

1 KA ¼ log n 1 þ KE

ðA44Þ

The mid-point slope is: 

dp dx

 ¼ x¼x50

KE n  ln10 a  n  ln10 ¼ 4  ð1 þ KE Þ 4

ðA45Þ

The point of inflection matches the mid-point (symmetric curve). xI ¼ x50

ðA46Þ

ðA48Þ

If we make the transformation x = log[A], the maximum value of p and the location parameter for the p/x curve are: a ¼ lim p ¼

This is equivalent to the Hill equation (Eq. (A1)) with the slope parameter m = n (the molecularity of the binding process). Analogously with the previous case, the geometric parameters are obtained by simple substitution. The upper asymptote is:

ðA47Þ

where k + 1 and k + 2 are association rate constants for agonist binding, k  1 and k  2 are dissociation rate constants for agonist unbinding, and a and b are the rate constants for the shutting and opening of the fully occupied channel. At this stage, it is worthwhile to distinguish between microscopic and macroscopic constants. The microscopic equilibrium dissociation constants for the first and second binding processes are KA1 = (k  1/k + 1), and KA2 = (k  2/ k + 2), whereas the equilibrium constant for the opening reaction is KE = b/a. The macroscopic equilibrium dissociation constants are (KA1/2), and 2KA2 for the first and second binding processes, respectively. The proportion of open channels at equilibrium is: p¼

n KE 1þKE ½A n KA 1þKE þ ½A

2kþ1

2A þ R] A þ AR] A2 R] A2 R*

The point of inflection matches the mid-point (symmetric curve). xI ¼ x50

41

KE 1 þ KE

ðA49Þ

and 0 x50 ¼ log@

K A2 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 K2A2 þ ð1 þ KE Þ  KA1 KA2 A 1 þ KE

ðA50Þ

The slope of the p/x curve is given by: dp 2  100x KE KA2 ð10x þ KA1 Þ  ln10 ¼ dx ð100x ð1 þ KE Þ þ KA ð2  10x þ KA ÞÞ2 2

ðA51Þ

The expression for the mid-point slope is too complicated to be given here explicitly. However, it would be worthwhile to comment on some points. The mid-point slope depends on the KE value, increasing as KE increases. The mid-point slope does not depend on the absolute values of KA1 and KA2, but on their ratio. If we define R = (KA1/ KA2), it is found that the mid-point slope increases as R increases. To analyze the (a)symmetry of the p/x curve, we need to obtain the second derivative of p with respect to x. Now, again, the resulting equation is too long to be written. However, it can be found that (d2p/dx2)x50 6¼ 0, and,

42

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

consequently, the curve is asymmetric for any combination of the KA1, KA2, and KE parameters. In addition, the second derivative approaches 0 for high values of KE, and this trend is faster as R increases. A.7. The operational model of agonism: the general case The operational model of agonism (Black & Leff, 1983) derives the E/[A] relationship starting from the first step of agonist action: the formation of the agonistreceptor complex. KA

A þ RU AR

By substituting Eq. (A54) into Eq. (A55), the following E/[A] relationship is obtained: E¼

½R0  ¼ ½R þ ½AR

ðA53Þ

ðA56Þ

where t = R0/KE. The E/x function is derived from Eq. (A56) by defining x= log[A]. E¼

Em  n 10nx ðKA þ 10x Þn þ  n 10nx

ðA57Þ

The upper asymptote is:

ðA52Þ

where KA is the agonist-receptor dissociation constant, KA = [A][R]/[AR]. It is assumed that the law of mass action is fulfilled.

Em  n ½An ðKA þ ½AÞn þ  n ½An

Em  n ðA58Þ x!1 1 þ n The mid-point, x50, is found by putting E = a/2 in Eq. (A57) and solving the equation for x. a ¼ lim E ¼

x50 ¼ log

KA

ðA59Þ

ð2 þ  n Þ1=n 1

where [R0] is the total receptor concentration. The fractional receptor occupancy is obtained by expressing [R] in terms of [AR] in the expression of KA and substituting in Eq. (A53).

To find the mid-point slope, we derive E with respect to x and substitute x with x50 in the resulting relationship:

½AR ½A ¼ ½R0  KA þ ½A

dE 10nx Em KA ð10x þ KA Þn1 n   n ln10 ¼ dx ðð10x þ KA Þn þ 10nx  n Þ2

ðA54Þ

The following logistic function is proposed as the transducer function of receptor occupancy into response. Em ½ARn E¼ n KE þ ½ARn



dE dx



¼

ðA60Þ

0:576  Em  n n  ð2 þ  n Þ  ðð2 þ  n Þ =n 1Þ

x¼x50

ð2 þ  n Þ =n ð1 þ  n Þ2

ðA55Þ

ðA61Þ

where Em is the maximum possible effect in a system and KE is the value of [AR] for half Em.

To determine the point of inflection, we derive Eq. (A60) with respect to x and set the resulting expression equal to 0.

d2 E 10nx n  Em KA  n un2 ð10x un þ n  KA un 10xþnx tn 10nx n  KA  n Þ  ln100 ¼ dx2 ðun þ 10nx tn Þ3 with u = 10x + KA. To make Eq. (A62) equal to 0, it is necessary that: 10x un þn  KA un 10xþnx  n 10nx n  KA  n ¼ 0 ðA63Þ Eq. (A63) can be solved easily for n = 1. After some algebra, we can write: x

x

x

ð10 þ KA Þ  ð10 KA þ 10 Þ ¼ 0

ðA64Þ

There are two possible solutions: 8 x < 10 þ KA ¼ 0 :

10x KA þ 10x  ¼ 0

ðA62Þ

The first solution is not possible because it implies that xI= log(  KA), and the log function is not defined for negative numbers. The second equation is the right one, and we find that the point of inflection for n = 1 is:   KA xI ¼ log ðA66Þ  þ1 Comparing Eq. (A66) with Eq. (A59), we see that the point of inflection matches the mid-point, and, consequently, the theoretical curve is symmetric for n = 1. A.8. The operational model of agonism: signal transduction by G-protein-coupled receptors

ðA65Þ

The operational model of agonism was applied to the case of signal transduction by GPCRs (Black & Leff, 1983).

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

Here, to produce an effect, a ternary complex agonistreceptor-G-protein must be formed. KA

KAR

A þ RU AR þ GUARG ½A  ½R ½AR  ½G ; KAR ¼ KA ¼ ½AR ½ARG

where KA is the agonist-receptor dissociation constant and KAR is the ternary complex dissociation constant. We will consider two limiting cases, (1) total receptor concentration greatly exceeds total G-protein concentration and, conversely, (2) total G-protein concentration greatly exceeds total receptor concentration, and the general case, (3) where no simplifying assumption is made between receptor and G-protein ratio. (1) [R0] >> [G0] The conservation of receptor concentration leads to: ½R0  ¼ ½R þ ½AR þ ½ARG ½R þ ½AR ðA68Þ and, thus ½R   ½A ðA69Þ ½AR ¼ 0 KA þ ½A The conservation of G-protein concentration leads to: ½G0  ¼ ½G þ ½ARG and, thus ½ARG ¼

½G0   ½AR KAR þ ½AR

ðA70Þ

ðA71Þ

By substituting Eq. (A69) into Eq. (A71), we have: ½ARG ¼

½G0   ½R0   ½A KA KAR þ ð½R0  þ KAR Þ  ½A

ðA72Þ

An [ARG]/x function: ½ARG ¼

a 1 þ 10xb x

ðA73Þ

a¼

ðA74Þ

The conservation of G-protein concentration leads to: ½G0  ¼ ½G þ ½ARG ½G

ðA77Þ

and, thus: ½ARG ¼

½G0   ½AR ½G0   ½R0   ½A KAR þ ½AR KA KAR þ ð½G0  þ KAR Þ  ½A ðA78Þ

An [ARG]/x function: ½ARG ¼

a 1 þ 10xb x

ðA79Þ

is derived from Eq. (A78) by defining: x ¼ log½A;

a¼

xb ¼ logb; where b ¼

½G0   ½R0  ½G0  þ KAR

KA KAR ; ½G0  þ KAR ðA80Þ

This [ARG]/x function follows a Hill equation (symmetric) with m = 1. The geometric parameters can be easily obtained by comparing Eq. (A79) with Eq. (A1). (3) [G0] [R0] When the total concentration of receptors or G-proteins are not negligible one relative to the other, the following quadratic equation is obtained (Black & Shankley, 1990; Jenkinson, 1989): ðA81Þ

y ¼ ½ARG;     KA b ¼  KAR  1 þ þ ½R0  þ ½G0  and c ¼ ½G   ½R  0 ½A 0 ðA82Þ

The [ARG]/x function follows a Hill equation (symmetric) with m = 1. The geometric parameters can be easily obtained by comparing Eq. (A73) with Eq. (A1). (2) [G0] >> [Ro] The conservation of receptor concentration leads to:

½R0  ¼ ½R þ ½AR þ ½ARG

ðA76Þ

where,

KA KAR xb ¼ logb; where b ¼ ; ½R0  þ KAR

½G0   ½R0  ½R0  þ KAR

½R0   KAR ½A KA KAR þ ðKAR þ ½GÞ  ½A

y2 þ b  y þ c ¼ 0

is derived from Eq. (A72) by defining: x ¼ log½A;

and, thus: ½AR ¼

ðA67Þ

43

ðA75Þ

The solution to Eq. (A81) is: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b2 4  c y¼ 2

ðA83Þ

We have not taken the positive square root in Eq. (A83) because only for the negative root, the condition [ARG] < [R0] and [ARG] < [G0], comes true.

44

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45

The upper asymptote of the [ARG]/[A] function is:

ð

1  K þ ½R0  þ ½G0  AR 2 ½A!1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  4  ½R0   ½G0  þ ðKAR þ ½R0  þ ½G0 Þ2

a ¼ lim ½ARG ¼

Þ ðA84Þ

The [A] giving 50% of the asymptote is: ½A50  ¼ 

ð

KA KAR  K

AR

þ ½R0  þ ½G0   3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4  ½R0   ½G0  þ ðKAR þ ½R0  þ ½G0 Þ2

Þ

2  K2AR þ 2  ½R0 2 5  ½R0   ½G0  þ 2  ½G0 2 þ 4  KAR ð½R0  þ ½G0 Þ

ðA85Þ The limit of [A50] as [R0] approaches 0 is normally considered the apparent agonist-receptor dissociation constant. lim ½A50  ¼

½R0 !0

KA ½G 

1 þ KAR0

ðA86Þ

The expressions for the derivatives of [ARG] are very long. Because of this, we show only the first derivative: d½ARG dx

  ¼ 21 10x  KA KAR þ 10x ðKAR þ ½R0  þ ½G0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  41þx 25x ½R0 ½G0  þ ðKA KAR þ 10x ðKAR þ ½R0  þ ½G0 ÞÞ2 ln10 þ10x ðKAR þ½R0  þ ½G0 Þln10  ð10x ð4  10x ½R0 ½G0 ln10 þðKAR ½R0 ½G0 ÞðKA KAR þ10x ðKAR ½R0  þ ½G0 ÞÞln10ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 41þx 25x ½R0 ½G0  þ ðKA KAR þ 10x ðKAR þ ½R0  þ ½G0 ÞÞ2

(A87) References Agneter, E., Singer, E. A., Sauermann, W., & Feuerstein, T. J. (1997). The slope parameter of concentration-response curves used as a touchstone for the existence of spare receptors. Naunyn Schmiedebergs Arch Pharmacol 356, 283 – 292. Agneter, E., Singer, E. A., Sauermann, W., & Feuerstein, T. J. (1998). Agneter et al. reply. Trends Pharmacol Sci 19, 445 – 446. Armstrong, D., & Strange, P. G. (2001). Dopamine D2 receptor dimer formation: evidence from ligand binding. J Biol Chem 276, 22621 – 22629. Bates, D. M., & Watts, D. G. (1988). Nonlinear Regression Analysis and Its Applications. New York: John Wiley and Sons. Black, J. (1996). A personal view of pharmacology. Annu Rev Pharmacol Toxicol 36, 1 – 33. Black, J. W., & Leff, P. (1983). Operational models of pharmacological agonism. Proc R Soc Lond B Biol Sci 220, 141 – 162. Black, J. W., & Shankley, N. P. (1990). Interpretation of agonist affinity estimations: the question of distributed receptor states. Proc R Soc Lond B Biol Sci 240, 503 – 518. Black, J. W., Leff, P., Shankley, N. P., & Wood, J. (1985). An operational model of pharmacological agonism: the effect of E/[A] curve shape on agonist dissociation constant estimation. Br J Pharmacol 84, 561 – 571. Boeynaems, J. M. & Dumont, J. E. (1980). Outlines of Receptor Theory. Amsterdam: Elsevier/North-Holland.

Bouvier, M. (2001). Oligomerization of G-protein-coupled transmitter receptors. Nat Rev Neurosci 2, 274 – 286. Chidiac, P., Green, M. A., Pawagi, A. B., & Wells, J. W. (1997). Cardiac muscarinic receptors. Cooperativity as the basis for multiple states of affinity. Biochemistry 36, 7361 – 7379. Christopoulos, A., & Lew, M. J. (2001). Beyond eyeballing: fitting models to experimental data. In: A. Christopoulos (Ed.), Biomedical Applications of Computer Modeling ( pp. 195 – 231). Boca Raton: CRC Press. Colquhoun, D. (1987). Affinity, efficacy, and receptor classification: is the classical theory still useful? In: J. W. Black, D. H. Gerskowitch, & D. H. Jenkinson (Eds.), Perspectives on Receptor Classification (pp. 103 – 114). New York: Alan R. Liss Inc. Colquhoun, D. (1998). Binding, gating, affinity and efficacy: the interpretation of structure-activity relationships for agonists and of the effects of mutating receptors. Br J Pharmacol 125, 924 – 947. del Castillo, J., & Katz, B. (1957). Interaction at end-plate receptors between different choline derivatives. Proc R Soc Lond B 146, 369 – 381. De Lean, A., Stadel, J. M., & Lefkowitz, R. J. (1980). A ternary complex model explains the agonist-specific binding properties of the adenylate cyclase-coupled b-adrenergic receptor. J Biol Chem 255, 7108 – 7117. Edsall, J. T., & Gutfreund, H. (1983). Biothermodynamics. The Study of Biochemical Processes at Equilibrium. Chichester: John Wiley and Sons. Feuerstein, T. J., & Limberger, N. (1999). Mathematical analysis of the control of neurotransmitter release by presynaptic receptors as a supplement to experimental data. Naunyn Schmiedebergs Arch Pharmacol 359, 345 – 359. Freund, R. J., & Litell, R. C. (Eds.). (1991). Nonlinear Models. In SAS System for Regression. Cary: SAS Institute. Furchgott, R. F. (1966). The use of b-haloalkylamines in the differentiation of receptors and in the determination of dissociation constants of receptor-agonist complexes. In: N. J. Harper, & A. B. Simmonds (Eds.), Advances in Drug Research (pp. 21 – 55). New York: Academic Press. Gether, U. (2000). Uncovering molecular mechanisms involved in activation of G protein-coupled receptors. Endocr Rev 21, 90 – 113. Giraldo, J. (1998). The slope parameter and the receptor reserve. Trends Pharmacol Sci 19, 445. Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality. Philos Trans R Soc Lond 36, 513 – 585. Green, M. A., Chidiac, P., & Wells, J. W. (1997). Cardiac muscarinic receptors. Relationship between the G protein and multiple states of affinity. Biochemistry 36, 7380 – 7394. Henis, Y. I., & Levitzki, A. (1979). Ligand competition curves as a diagnostic tool for delineating the nature of site-site interactions: theory. Eur J Biochem 102, 449 – 465. Hey, C., Wessler, I., & Racke, K. (1994). Muscarinic inhibition of endogenous noradrenaline release from rabbit isolated trachea: receptor subtype and receptor reserve. Naunyn Schmiedebergs Arch Pharmacol 350, 464 – 472. Jenkinson, D. H. (1989). Finding affinity constants for agonists by the irreversible antagonists method. Trends Pharmacol Sci 10, 16 – 17. Kenakin, T. (1997a). Differences between natural and recombinant G protein-coupled receptor systems with varying receptor/G protein stoichiometry. Trends Pharmacol Sci 18, 456 – 464. Kenakin, T. (1997b). Pharmacologic Analysis of Drug-Receptor Interaction (3rd edn.). Philadelphia: Lippincott-Raven Publishers. Kenakin, T. (1998). Multiple receptor conformations and agonism. Trends Pharmacol Sci 19, 210 – 211. Klotz, I. M., & Hunston, D. L. (1979). Protein affinities for small molecules: conceptions and misconceptions. Arch Biochem Biophys 193, 314 – 328. Lee, T. W., Sole, M. J., & Wells, J. W. (1986). Assessment of a ternary model for the binding of agonists to neurohumoral receptors. Biochemistry 25, 7009 – 7020. Leff, P., Prentice, D. J., Giles, H., Martin, G. R., & Wood, J. (1990). Esti-

J. Giraldo et al. / Pharmacology & Therapeutics 95 (2002) 21–45 mation of agonist affinity and efficacy by direct, operational modelfitting. J Pharmacol Methods 23, 225 – 237. Leff, P., Scaramellini, C., Law, C., & McKechnie, K. (1997). A three-state receptor model of agonist action. Trends Pharmacol Sci 18, 355 – 362. Lefkowitz, R. J., Cotecchia, S., Samama, P., & Costa, T. (1993). Constitutive activity of receptors coupled to guanine nucleotide regulatory proteins. Trends Pharmacol Sci 14, 303 – 307. Lewis, T. M., Sivilotti, L. G., Colquhoun, D., Gardiner, R. M., Schoepfer, R., & Rees, M. (1998). Properties of human glycine receptors containing the hyperekplexia mutation a1(K276E), expressed in Xenopus oocytes. J Physiol 507, 25 – 40. Mackay, D. (1988). Concentration-response curves and receptor classification: null method or operational model? Trends Pharmacol Sci 9, 202 – 205. Milligan, G. (2001). Oligomerisation of G-protein-coupled receptors. J Cell Sci 114, 1265 – 1271. Pardo, L., Campillo, M., & Giraldo, J. (1997). The effect of the molecular mechanism of G protein-coupled receptor activation on the process of signal transduction. Eur J Pharmacol 335, 73 – 87. Richards, F. J. (1959). A flexible growth function for empirical use. J Exp Bot 10, 290 – 300. Samama, P., Cotecchia, S., Costa, T., & Lefkowitz, R. J. (1993). A mutation-induced activated state of the b2-adrenergic receptor. Extending the ternary complex model. J Biol Chem 268, 4625 – 4636. Sips, S. (1948). On the structure of a catalyst surface. J Chem Phys 16, 490 – 495. Sips, S. (1950). On the structure of a catalyst surface. II. J Chem Phys 18, 1024 – 1026. Stephenson, R. P. (1956). A modification of receptor theory. Br J Pharmacol 11, 379 – 393. Strange, P. G. (1999). G-protein coupled receptors: conformations and states. Biochem Pharmacol 58, 1081 – 1088.

45

Tabernero, A., Giraldo, J., & Vila, E. (1999). Modelling the changes due to the endothelium and hypertension in the a-adrenoreceptor-mediated responses of rat aorta. J Auton Pharmacol 19, 219 – 228. Tallman, J. (2000). Dimerization of G-protein-coupled receptors: implications for drug design and signaling. Neuropsychopharmacology 23, S1 – S2. Taylor, P., & Insel, P. A. (1990). Molecular basis of pharmacologic selectivity. In: W. B. Prat, & P. Taylor (Eds.), Principles of Drug Action. The Basis of Pharmacology (pp. 1 – 102). New York: Churchill Livingstone. Van Der Graaf, P. H., & Schoemaker, R. C. (1999). Analysis of asymmetry of agonist concentration-effect curves. J Pharmacol Toxicol Methods 41, 107 – 115. Vivas, N. M., Giraldo, J., Tabernero, A., Vila, E., & Badia, A. (1997). Use of the operational model of agonism and [3H]prazosin binding to assess altered responsiveness of a1-adrenoceptors in the vas deferens of spontaneously hypertensive rat. Naunyn Schmiedebergs Arch Pharmacol 356, 383 – 391. Weiss, J. M., Morgan, P. H., Lutz, M. W., & Kenakin, T. P. (1996a). The cubic ternary complex receptor-occupancy model. I. Model description. J Theor Biol 178, 151 – 167. Weiss, J. M., Morgan, P. H., Lutz, M. W., & Kenakin, T. P. (1996b). The cubic ternary complex receptor-occupancy model. II. Understanding apparent affinity. J Theor Biol 178, 169 – 182. Weiss, J. M., Morgan, P. H., Lutz, M. W., & Kenakin, T. P. (1996c). The cubic ternary complex receptor-occupancy model. III. Resurrecting efficacy. J Theor Biol 181, 381 – 397. Wells, J. W. (1992). Analysis and interpretation of binding at equilibrium. In: E. C. Hulme (Ed.), Receptor-Ligand Interactions. A Practical Approach (pp. 289 – 395). Oxford: Oxford University Press. Wyman, J., & Gill, S. J. (1990). Binding and Linkage. Functional Chemistry of Biological Macromolecules. Mill Valley: University Science Books.